7.6-bo'lim : Kvadratlarni o'z ichiga olgan integrallar
1. \( \displaystyle \int{{\frac{7}{{{w^2} + 3w + 3}}\,dw}}\) integralini baholang.
Yechim
Birinchi narsa qilish kerak bo'lgan kvadratni maxrajdagi kvadratchada to'ldirish (kvadrat tafsilotlarini to'ldirishni tekshirishni sizga qoldiramiz).
\[\int{{\frac{7}{{{w^2} + 3w + 3}}\,dw}} = \int{{\frac{7}{{{{\left( {w + \frac{3}{2}} \right)}^2} + \frac{3}{4}}}\,dw}}\]2-qadam
Bundan ko'rishimiz mumkinki, quyidagi almashtirish biz uchun ishlashi kerak.
\[u = w + \frac{3}{2}\hspace{0.5in} \Rightarrow \hspace{0.5in}du = dw\]O'zgartirishni amalga oshirish beradi,
\[\int{{\frac{7}{{{w^2} + 3w + 3}}\,dw}} = \int{{\frac{7}{{{u^2} + \frac{3}{4}}}\,du}}\]3-qadam
Ushbu integral ushbu bo'lim boshida berilgan formula bilan amalga oshirilishi mumkin.
\[\int{{\frac{7}{{{w^2} + 3w + 3}}\,dw}} = \frac{{14}}{{\sqrt 3 }}{\tan ^{ - 1}}\left( {\frac{{2u}}{{\sqrt 3 }}} \right) + c = \require{bbox} \bbox[2pt,border:1px solid black]{{\frac{{14}}{{\sqrt 3 }}{{\tan }^{ - 1}}\left( {\frac{{2w + 3}}{{\sqrt 3 }}} \right) + c}}\]\(u\) o'rniga qayta yozishni unutmang!